Let $n \geq k$. We say that a permutation $\sigma \in S_n$ contains a permutation (or "pattern") $\tau \in S_k$ if there is a substring of $k$ (not necessarily consecutive) elements of $\sigma$ ordered like $\tau$. For example, the permutation $\sigma=24513$ contains $132$, since the string $243$ in $\sigma$ has the same relative order as $132$.
Similarly, it contains $123$ (the string $245$), $213$ (the string $213$), $231$ (the string $451$) and $312$ (the string $513$). The only length-$3$ pattern it does not contain is $321$.
Question: Given a permutation $\tau \in S_k$, is it possible to find a value of $n$ and a permutation $\sigma \in S_n$ such that $\sigma$ contains every pattern of length $k$ except $\tau$?
Overly Ambitious Generalization: Given a permutation $\sigma \in S_n$ and a $k \leq n$, let $T_k(\sigma)$ be the set of patterns of length $k$ contained in $\sigma$. For a fixed integer $k$, what sets of patterns can possibly appear as $T_k(\sigma)$ for some $\sigma$?
For large $k$, almost all subsets do not ever occur as a $T_k(\sigma)$. One way to see this: By Erdős–Szekeres every permutation with $n>(k-1)^2$ contains either $123\dots k$ or $k\dots 321$. There's less than $(k^2)!$ other choices for $\sigma$, each of which contains at most $\binom{k^2}{k}$ patterns. So at most $\binom{k^2}{k} (k^2)!$ subsets containing neither $123\dots k$ nor $k \dots 321$ ever appear as a $T_k$, which is much smaller than the total number of subsets not containing those two patterns.
The first question arose out of a discussion I had with a computer science graduate student about the complexity of testing for pattern containment. Most of the work I've seen on pattern-avoiding permutations has focused more on enumerating them, and I'd appreciate any references to other work done on relationships between various pattern containments.
